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If x and y are positive integers, what is the greatest

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Re: If x and y are positive integers, what is the greatest [#permalink]

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New post 08 Jul 2014, 00:37
If 5x and 3y are consecutive, does that mean x and y will always b consecutive too?

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New post 08 Jul 2014, 06:27
out of all the questions which I have encountered in my GMAT prep , this is the scariest.. Mind you not the toughest but still the scariest

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Bunuel wrote:
carcass wrote:
If x and y are positive integers, what is the greatest common divisor of x and y?

1) 2x + y = 73
2) 5x – 3y = 1

MMMMMMmm

Here I'm not sure that the answer is C because is true that we need of both statement to find possible values for X and Y. Infact statement 1 and 2 we do not have values for the variables (can be everything).

But it seems to be a trap answer......


If x and y are positive integers, what is the greatest common divisor of x and y?

This is a classic "C trap" question: "C trap" is a problem which is VERY OBVIOUSLY sufficient if both statements are taken together. When you see such question you should be extremely cautious when choosing C for an answer.

(1) \(2x+y=73\). Suppose GCD(x, y) is some integer \(d\), then \(x=md\) and \(y=nd\), for some positive integers \(m\) and \(n\). So, we'll have \(2(md)+(nd)=d(2m+n)=73\). Now, since 73 is a prime number (73=1*73) then \(d=1\) and \(2m+n=73\) (vice versa is not possible because \(m\) and \(n\) are positve integers and therefore \(2m+n\) cannot equal to 1). Hence we have that GCD(x, y)=d=1. Sufficient.

(2) \(5x-3y=1\) --> \(5x=3y+1\) --> \(5x\) and \(3y\) are consecutive integers. Two consecutive integers are co-prime, which means that they don't share ANY common factor but 1 (for example 20 and 21 are consecutive integers, thus only common factor they share is 1). So, \(5x\) and \(3y\) don't share any common factor but 1, thus \(x\) and \(y\) also don't share any common factor but 1. Hence, GCD(x, y) is 1. Sufficient.

Answer: D.

Hope it's clear.


Thanks for your exp, Bunuel. That's awesome!!! :shock: :shock: :shock: :shock: :shock:
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New post 18 Aug 2014, 10:57
Bunuel wrote:
\(5x-3y=1\) --> \(5x=3y+1\) --> \(5x\) and \(3y\) are consecutive integers. Two consecutive integers are co-prime, which means that they don't share ANY common factor but 1 (for example 20 and 21 are consecutive integers, thus only common factor they share is 1). So, \(5x\) and \(3y\) don't share any common factor but 1, thus \(x\) and \(y\) also don't share any common factor but 1. Hence, GCD(x, y) is 1. Sufficient.


Hope it's clear.

Hi Bunuel,
Although 5x and 3y are consecutive integers and co-prime , why are x and y co-prime?
is it because 5 and 3 are also co-prime?

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New post 18 Aug 2014, 11:21
ronr34 wrote:
Bunuel wrote:
\(5x-3y=1\) --> \(5x=3y+1\) --> \(5x\) and \(3y\) are consecutive integers. Two consecutive integers are co-prime, which means that they don't share ANY common factor but 1 (for example 20 and 21 are consecutive integers, thus only common factor they share is 1). So, \(5x\) and \(3y\) don't share any common factor but 1, thus \(x\) and \(y\) also don't share any common factor but 1. Hence, GCD(x, y) is 1. Sufficient.


Hope it's clear.

Hi Bunuel,
Although 5x and 3y are consecutive integers and co-prime , why are x and y co-prime?
is it because 5 and 3 are also co-prime?


Let me ask you a question: if x and y shared any common factor but 1, would 5x and 3y be co-prime? Wouldn't they also share that factor?
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New post 29 Sep 2014, 20:23
Hi Bunuel
You said GCD(x,y)= d then the relation between x and y is that, x=md and y=nd

What about LCM?

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New post 19 Mar 2015, 11:00
Bunuel wrote:
So, \(5x\) and \(3y\) don't share any common factor but 1, thus \(x\) and \(y\) also don't share any common factor but 1. Hence, GCD(x, y) is 1. Sufficient.



Can someone break this down for me? How do we know that because 5x and 3y don't share any common factors other than 1, x and y also won't share any common factors but 1? Is it because 5 and 3 do not share any common factors?

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New post 19 Mar 2015, 19:49
deerhunter wrote:
Bunuel wrote:
So, \(5x\) and \(3y\) don't share any common factor but 1, thus \(x\) and \(y\) also don't share any common factor but 1. Hence, GCD(x, y) is 1. Sufficient.



Can someone break this down for me? How do we know that because 5x and 3y don't share any common factors other than 1, x and y also won't share any common factors but 1? Is it because 5 and 3 do not share any common factors?


Think of it this way - say, x and y shared a common factor 2. Then 2 would be a factor of 5x as well as 3y. But we are given that 5x and 3y share no common factor. Hence, x and y can share no common factor.
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If x and y are positive integers, what is the greatest common divisor of x and y?

1. 2x + y = 73
2. 5x – 3y = 1

Another way of solving it .Both x and y are integers .
From 1: x =(73-y)/2 .Since x is a integer it implies 73-y =even number .73 is Odd so y is also Odd .X is even so GCM will be 1.Sufficient

From 2 :5x-3y =1 .They are consecutive numbers i.e .odd-even or even -odd .so the GCM in this case =1 .Sufficient

Option D is correct .

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zest4mba wrote:
If x and y are positive integers, what is the greatest common divisor of x and y?

(1) 2x + y = 73
(2) 5x – 3y = 1


Question : GCD of x and y = ?

Statement 1: 2x + y = 73

This statement can give us multiple solutions of x and y but the important part is to notice the value of GCD in each case e.g.
(y=1, x=36) GCD = 1
(y=3, x=35) GCD = 1
(y=5, x=34) GCD = 1
(y=7, x=33) GCD = 1
(y=9, x=32) GCD = 1... and so on...

Finally we realize that instead of multiple solutions of x and y, their GCD is consistently 1,
Hence SUFFICIENT

Statement 2: 5x – 3y = 1

(y=3, x=2) GCD = 1
(y=8, x=5) GCD = 1
(y=13, x=8) GCD = 1
(y=18, x=11) GCD = 1
(y=23, x=14) GCD = 1... and so on...

Finally we realize that instead of multiple solutions of x and y, their GCD is consistently 1,
Hence SUFFICIENT

Answer: Option
[Reveal] Spoiler:
D


Point to Learn: In all such equations with two variable you can realize that the solutions have a harmony i.e. value of variable x changes by co-efficient of y and value of y changes by co-efficient of x and this relation holds true in all such equation where the GCD of co-efficients of x and y is 1.

If there is some common factor among co-efficients of x and y then cancel the common factor and the rule holds true in those cases with modified equation.


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New post 30 Jan 2016, 14:35
Banuel - I was hoping you could help me understand something.

If #1 had said 2x+y = 75 (not prime), what would using your equation tell me?

So I'd have d(2m + n) = 70

Factors of 70:
1, 70
2, 35
5, 14
7, 10

Following your logic, I could have:
d = 1 OR? 2
(2m + n) = 5, 7, 10, 14, 35, 70

or is 2 the GCD? Or can you not tell because of all the potential values of 2m + n?

Thank you soooo much ahead of time for taking the time to explain.

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New post 16 Mar 2016, 04:29
Here is what i did => i made the pairs of values and saw the pattern and then compiled that D is correct
still not able to see a proper solution on this page
some are quoting algebra and some are doing by values putting
maybe chetan2u will be helpful here..
Any other methods?
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Chiragjordan wrote:
Here is what i did => i made the pairs of values and saw the pattern and then compiled that D is correct
still not able to see a proper solution on this page
some are quoting algebra and some are doing by values putting
maybe chetan2u will be helpful here..
Any other methods?



Have you checked out Bunuel's solution on the first page?
if-x-and-y-are-positive-integers-what-is-the-greatest-128552.html#p1053416

It explains the best way to deal with this question.

If you want to avoid algebra, think about it like this:

If x and y are positive integers, what is the greatest common divisor of x and y?

1) 2x + y = 73
Say, x and y have a common factor f other than 1. If that is the case, you should be able to take f common out of the two terms on left hand side. So you will get
f*something = 73
But 73 cannot be written as product of two numbers other than 1 and itself. So f MUST BE 1. Hence greatest common divisor of x and y MUST BE 1. Sufficient

2) 5x – 3y = 1
Here, 5x and 3y are consecutive integers (since difference between them is 1). Consecutive integers can share no common factor other than 1. So 5x and 3y have no common factors. This means that x and y can have no common factors (other than 1) too. Else that factor would have been common between 5x and 3y too. Hence greatest common divisor of x and y MUST BE 1. Sufficient

Answer (D)
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New post 22 Apr 2016, 03:23
Bunuel wrote:
carcass wrote:
If x and y are positive integers, what is the greatest common divisor of x and y?

1) 2x + y = 73
2) 5x – 3y = 1

MMMMMMmm

Here I'm not sure that the answer is C because is true that we need of both statement to find possible values for X and Y. Infact statement 1 and 2 we do not have values for the variables (can be everything).

But it seems to be a trap answer......


If x and y are positive integers, what is the greatest common divisor of x and y?

This is a classic "C trap" question: "C trap" is a problem which is VERY OBVIOUSLY sufficient if both statements are taken together. When you see such question you should be extremely cautious when choosing C for an answer.

(1) \(2x+y=73\). Suppose GCD(x, y) is some integer \(d\), then \(x=md\) and \(y=nd\), for some positive integers \(m\) and \(n\). So, we'll have \(2(md)+(nd)=d(2m+n)=73\). Now, since 73 is a prime number (73=1*73) then \(d=1\) and \(2m+n=73\) (vice versa is not possible because \(m\) and \(n\) are positve integers and therefore \(2m+n\) cannot equal to 1). Hence we have that GCD(x, y)=d=1. Sufficient.

(2) \(5x-3y=1\) --> \(5x=3y+1\) --> \(5x\) and \(3y\) are consecutive integers. Two consecutive integers are co-prime, which means that they don't share ANY common factor but 1 (for example 20 and 21 are consecutive integers, thus only common factor they share is 1). So, \(5x\) and \(3y\) don't share any common factor but 1, thus \(x\) and \(y\) also don't share any common factor but 1. Hence, GCD(x, y) is 1. Sufficient.

Answer: D.

Hope it's clear.


Hi Bunuel,

Please help me understand - is it that even if two numbers' multiples are co primes, that the numbers themselves will be co primes as well? How? Thank you.

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New post 22 Apr 2016, 03:27
abypatra wrote:
Bunuel wrote:
carcass wrote:
If x and y are positive integers, what is the greatest common divisor of x and y?

1) 2x + y = 73
2) 5x – 3y = 1

MMMMMMmm

Here I'm not sure that the answer is C because is true that we need of both statement to find possible values for X and Y. Infact statement 1 and 2 we do not have values for the variables (can be everything).

But it seems to be a trap answer......


If x and y are positive integers, what is the greatest common divisor of x and y?

This is a classic "C trap" question: "C trap" is a problem which is VERY OBVIOUSLY sufficient if both statements are taken together. When you see such question you should be extremely cautious when choosing C for an answer.

(1) \(2x+y=73\). Suppose GCD(x, y) is some integer \(d\), then \(x=md\) and \(y=nd\), for some positive integers \(m\) and \(n\). So, we'll have \(2(md)+(nd)=d(2m+n)=73\). Now, since 73 is a prime number (73=1*73) then \(d=1\) and \(2m+n=73\) (vice versa is not possible because \(m\) and \(n\) are positve integers and therefore \(2m+n\) cannot equal to 1). Hence we have that GCD(x, y)=d=1. Sufficient.

(2) \(5x-3y=1\) --> \(5x=3y+1\) --> \(5x\) and \(3y\) are consecutive integers. Two consecutive integers are co-prime, which means that they don't share ANY common factor but 1 (for example 20 and 21 are consecutive integers, thus only common factor they share is 1). So, \(5x\) and \(3y\) don't share any common factor but 1, thus \(x\) and \(y\) also don't share any common factor but 1. Hence, GCD(x, y) is 1. Sufficient.

Answer: D.

Hope it's clear.


Hi Bunuel,

Please help me understand - is it that even if two numbers' multiples are co primes, that the numbers themselves will be co primes as well? How? Thank you.


Can you please give an example of what you mean?
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New post 22 Apr 2016, 03:55
This is a classic "C trap" question: "C trap" is a problem which is VERY OBVIOUSLY sufficient if both statements are taken together. When you see such question you should be extremely cautious when choosing C for an answer.

(1) \(2x+y=73\). Suppose GCD(x, y) is some integer \(d\), then \(x=md\) and \(y=nd\), for some positive integers \(m\) and \(n\). So, we'll have \(2(md)+(nd)=d(2m+n)=73\). Now, since 73 is a prime number (73=1*73) then \(d=1\) and \(2m+n=73\) (vice versa is not possible because \(m\) and \(n\) are positve integers and therefore \(2m+n\) cannot equal to 1). Hence we have that GCD(x, y)=d=1. Sufficient.

(2) \(5x-3y=1\) --> \(5x=3y+1\) --> \(5x\) and \(3y\) are consecutive integers. Two consecutive integers are co-prime, which means that they don't share ANY common factor but 1 (for example 20 and 21 are consecutive integers, thus only common factor they share is 1). So, \(5x\) and \(3y\) don't share any common factor but 1, thus \(x\) and \(y\) also don't share any common factor but 1. Hence, GCD(x, y) is 1. Sufficient.

Answer: D.

Hope it's clear.[/quote]

Hi Bunuel,

Please help me understand - is it that even if two numbers' multiples are co primes, that the numbers themselves will be co primes as well? How? Thank you.[/quote]

Can you please give an example of what you mean?[/quote]

In the explanation of the second AC, you state that 5x and and 3y dont share any other common factor other than 1 (since they are co prime), and hence x and y would also be co prime. I tried the same with an example for x and y, 2 and 3 respectively and the equation holds true. But would 5x=3y+1 result in x and y as co prime for all values of 5x=3y+1?

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New post 22 Apr 2016, 04:01
abypatra wrote:

In the explanation of the second AC, you state that 5x and and 3y dont share any other common factor other than 1 (since they are co prime), and hence x and y would also be co prime. I tried the same with an example for x and y, 2 and 3 respectively and the equation holds true. But would 5x=3y+1 result in x and y as co prime for all values of 5x=3y+1?


Let me ask you a question: if x and y shared any common factor but 1, would 5x and 3y be co-prime? Wouldn't they also share that factor?
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If x and y are positive integers, what is the greatest [#permalink]

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New post 22 Apr 2016, 04:11
Bunuel wrote:
abypatra wrote:

In the explanation of the second AC, you state that 5x and and 3y dont share any other common factor other than 1 (since they are co prime), and hence x and y would also be co prime. I tried the same with an example for x and y, 2 and 3 respectively and the equation holds true. But would 5x=3y+1 result in x and y as co prime for all values of 5x=3y+1?


Let me ask you a question: if x and y shared any common factor but 1, would 5x and 3y be co-prime? Wouldn't they also share that factor?


No they would not be co primes then... Understood. Thank you for helping me understand :)

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If x and y are positive integers, what is the greatest   [#permalink] 22 Apr 2016, 04:11

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